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arXiv:1103.1150 · Physical Review A

Semigroup evolution in the Wigner-Weisskopf pole approximation with Markovian spectral coupling

F. Shikerman, Avi Pe’er, Lawrence Paul Horwitz

Published Jul 27, 2011• 0 citations
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Abstract

We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation, the evolution of a total system subspace is not an exact semigroup for multichannel decay unless the projectors into eigenstates of the reduced evolution generator $W(z)$ are orthogonal. With multichannel decay, the projectors must be evaluated at different pole locations ${z}_{\ensuremath{\alpha}}\ensuremath{\ne}{z}_{\ensuremath{\beta}}$, and since the orthogonality relation does not generally hold at different values of $z$, the semigroup evolution is a poor approximation for the multichannel decay, even for very weak coupling. Nevertheless, if the theory is generalized to take into account interactions with an environment, one can ensure orthogonality of the $W(z)$ projectors regardless of the number of poles. Such a possibility occurs when $W(z)$, and hence its eigenvectors, is independent of $z$, which corresponds to the Markovian limit of the coupling to the continuum spectrum.

DOI
10.1103/physreva.84.012122

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We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation, the evolution of a total system subspace is not an exact semigroup for multichannel decay unless the projectors into eigenstates of the reduced evolution generator W(z)W(z)W(z) are orthogonal. With multichannel decay, the projectors must be evaluated at different pole locations z\ensuremathα\ensuremath≠z\ensuremathβ{z}_{\ensuremath{\alpha}}\ensuremath{\ne}{z}_{\ensuremath{\beta}}z\ensuremathα​\ensuremath=z\ensuremathβ​, and since the orthogonality relation does not generally hold at different values of zzz, the semigroup evolution is a poor approximation for the multichannel decay, even for very weak coupling. Nevertheless, if the theory is generalized to take into account interactions with an environment, one can ensure orthogonality of the W(z)W(z)W(z) projectors regardless of the number of poles. Such a possibility occurs when W(z)W(z)W(z), and hence its eigenvectors, is independent of zzz, which corresponds to the Markovian limit of the coupling to the continuum spectrum.
The description of an open quantum system's decay almost always requires several approximations so as to remain tractable. In this Letter, we first revisit the meaning, domain, and seeming contradictions of a few of the most widely used of such approximations: (semigroup) Markovianity, linear response theory, Wigner-Weisskopf approximation, and rotating-wave approximation. Second, we derive an effective time-dependent decay theory and corresponding generalized quantum regression relations for an open quantum system linearly coupled to an environment. This theory covers all timescales and subsumes the Markovian and linear-response results as limiting cases. Finally, we apply our theory to the phenomenon of quantum friction.
BEYOND THE WEISSKOPF–WIGNER APPROXIMATION IN THE K0−Kˉ0K^0-\bar K^0K0−Kˉ0 SYSTEM
We reanalyze the time evolution of the [Formula: see text] system in the language of certain spectral function whose Fourier transforms give the time-dependent survival and transition amplitudes. Approximating the spectral function by a one-pole ansatz we obtain insight into the limitation of the validity of the one-pole approximation. It will be shown that the same validity restrictions apply to the known formulae of the Weisskopf–Wigner approximation as well. The present analysis can also be applied to the effect of so-called vacuum regeneration of KL and KS. As a result of this possibility new contributions to the well-known oscillatory terms will enter the time-dependent transition probabilities. It will be shown that the order of magnitude of this new effect is very small and, in principle, its exact determination lies outside the scope of the one-pole ansatz.
In this talk we discuss how to combine aspects of the theory of stable relativistic particles and of the theory of non-relativistic decaying particles into a relativistic theory of unstable particles and resonances. For the theory of stable relativistic particles we use Wigner’s description by unitary irreducible representations (UIR) of the Poincaré group [ 1 ] which has been the foundation for relativistic quantum mechanics and field theory [ 2 ]. For the non-relativistic decaying particles we use a time asymmetric quantum mechanics in which the Hilbert space axiom, that allows only symmetric (unitary group) time evolution, is replaced by time asymmetric boundary conditions which lead to an irreversible , semigroup time evolution [ 3 ]. In the limit in which certain contributions to the energy continuum are neglected, this new quantum mechanics leads to the Weisskopf-Wigner approximate methods [ 4 ] for decay, and to the Breit-Wigner energy distribution [ 5 ] for the resonance lineshape. We thus generalize and combine two fundamental contributions of Wigner into a description of relativistic resonances and decaying states. The central concept of this description is the relativistic Gamow vector, which is a generalized eigenvector (ket) of the self-adjoint invariant square mass operator https://www.w3.org/1998/Math/MathML" display="inline"> M 2 = P μ P μ P μ = P 1 μ + P 2 μ in resonance scattering 1±2→ R →3±4) with a complex eigenvalue, a relativistic Breit-Wigner energy distribution, and an exponential time evolution governed by a semigroup of continuous operators in the space of kets.
Basic notions and theWigner–Weisskopf theory Although the notions of bound states, scattering and quantum transitions are well defined in the quantum theory, the description of an unstable system, involved in the process of decay, has remained an outstanding issue for many years. The problem is fundamental, since it concerns the nature of irreversible processes, one of the most important issues in statistical mechanics and the theme that is central to this book. The theory of decay is intimately connected with scattering theory and necessarily contains mathematical ideas and methods. We shall try to explain these points carefully as we get to them. We treat elsewhere in the book the ideas of Boltzmann, Van Hove and Prigogine on irreversible phenomena. The tools that are developed there are basically approximate, although very useful. One can argue that the basic rigorous characteristic of an irreversible process is that, as represented in terms of the evolution of a state in the Hilbert space of the quantum theory, it must be a semigroup. This type of evolution, resulting in an operation Z ( t ) on a state ψ , should satisfy the property Z ( t 2 ) Z ( t 1 ) = Z ( t 1 + t 2 ). (17.1) The argument is as follows. If the system evolves in time t 1 and is stopped, then evolves further at time t 2 , since the process has no memory, the total evolution should be as if the system evolved from the initial state to a state at t 1 + t 2 independently of the fact that it was done in two stages (Piron, 1976). Since the process is irreversible, the operator Z ( t ) may have no inverse. Such an evolution is called a semigroup . As we shall see, it is not possible to obtain such an evolution law in the framework of the standard quantum theory (Horwitz et al. , 1971), but recently much work has been done, and methods have been developed, based on ideas of Sz.-Nagy and Foias (1976), such as the theory of Lax and Phillips (1967) and its extension to the quantum theory (Strauss et al. , 2000) in which semigroup evolution can be achieved.
TIME EVOLUTION OF K0−K0ˉK^0-\bar{K^0}K0−K0ˉ SYSTEM IN SPECTRAL FORMULATION
We reanalyse the time evolution of the K0−K0ˉK^0-\bar{K^0}K0−K0ˉ system in the language of certain spectral function whose Fourier transforms give the time dependent survival and transition amplitudes. The reanalysis turned out to be necessary in view of the astonishing theorem by Khalfin on the possibility of vacuum regeneration of KSK_SKS​ and KLK_LKL​. The main reason for this unexpected behaviour is the non-orthogonality of $\ks$ and $\kl$. As a result of this theorem new contributions to the well known oscillatory terms will enter the time dependent transition probabilities. These new terms are not associated with small/large time behaviour of the amplitudes and therefore their magnitude is apriori unknown. Approximating the spectral functions by an one-pole ansatz Khalfin estimated the new effect in transition probabilities to be 4×10−44 \times 10^{-4}4×10−4. Whereas we agree with Khalfin on the general existence of vacuum regeneration of KSK_SKS​ and KLK_LKL​ we disagree on the size of the effect. A careful analysis of the one-pole approximation reveals that the effect is eleven orders of magnitude smaller than Khalfin's estimate and, in principle, its exact determination lies outside the scope of the one-pole ansatz. The present paper gives also insight into the limitation of the validity of one-pole approximation, not only for small/large time scales, but also for intermediate times where new effects, albeit small, are possible. It will be shown that the same validity restrictions apply to the known formulae of Weisskopf-Wigner approximation as well.
We reanalyse the time evolution of the K0 -`K0 system in the language of certain spectral function whose Fourier transforms give the time dependent survival and transition amplitudes. Approximating the spectral function by an one-pole ansatz we give insight into the limitation of the validity of one-pole approximation, not only for small/large time scales, but also for intermediate times where new effects, albeit small, are possible. It will be shown that the same validity restriction apply to the known formulae of Weisskopf-Wigner approximation as well. The present analysis can also be applied to the effect of vacuum regeneration of KL and KS, a possibility pointed out by Khalfin. As a result of this possible new contributions to the well known oscillatory terms will enter the time depend transition probanilities. These new terms are not associated with small/large time behaviour of the amplitudes and therefore their magnitude is apriori unknown. It will be shown that the order of magnitude of this new effect is very small and, in principle, its exact determination lies outside the scope of one-pole ansatz.